Embedded newform 441.2.bn.a.353.17

• Label: 441.2.bn.a.353.17
• Level: $441$
• Weight: $2$
• Character: 441.353
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bn (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			353.17
Character	\(\chi\)	\(=\)	441.353
Dual form			441.2.bn.a.5.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08970 - 0.869006i) q^{2} +(0.851533 - 1.50827i) q^{3} +(-0.0127693 - 0.0559462i) q^{4} +(-2.47373 + 1.68656i) q^{5} +(-2.23861 + 0.903578i) q^{6} +(-0.888042 - 2.49226i) q^{7} +(-1.24418 + 2.58356i) q^{8} +(-1.54978 - 2.56869i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08970	−	0.869006i	−0.770534	−	0.614480i	0.157267	−	0.987556i	\(-0.449732\pi\)
							−0.927801	+	0.373076i	\(0.878303\pi\)
\(3\)	0.851533	−	1.50827i	0.491633	−	0.870803i
							\(5\)	−2.47373	+	1.68656i	−1.10629	+	0.754253i	−0.971599	−	0.236635i	\(-0.923955\pi\)
−0.134687	+	0.990888i	\(0.543003\pi\)
\(7\)	−0.888042	−	2.49226i	−0.335648	−	0.941987i
							\(11\)	−0.0467177	+	0.309952i	−0.0140859	+	0.0934540i	−0.994769	−	0.102149i	\(-0.967428\pi\)
0.980683	+	0.195603i	\(0.0626663\pi\)
\(13\)	−0.257569	+	1.70886i	−0.0714369	+	0.473953i	0.924296	+	0.381675i	\(0.124653\pi\)
							−0.995733	+	0.0922774i	\(0.970585\pi\)
\(17\)	0.837659	+	0.777234i	0.203162	+	0.188507i	0.775192	−	0.631726i	\(-0.217653\pi\)
							−0.572030	+	0.820233i	\(0.693844\pi\)
\(19\)	−4.78224	+	2.76103i	−1.09712	+	0.633423i	−0.935463	−	0.353423i	\(-0.885018\pi\)
							−0.161658	+	0.986847i	\(0.551684\pi\)
\(23\)	−0.766163	+	2.48384i	−0.159756	+	0.517916i	−0.999693	−	0.0247672i	\(-0.992116\pi\)
							0.839937	+	0.542684i	\(0.182592\pi\)
\(29\)	−1.75791	+	1.89457i	−0.326435	+	0.351814i	−0.874851	−	0.484392i	\(-0.839041\pi\)
							0.548416	+	0.836206i	\(0.315231\pi\)
\(31\)		−	2.95542i		−	0.530809i	−0.964137	−	0.265404i	\(-0.914495\pi\)
							0.964137	−	0.265404i	\(-0.0855055\pi\)
\(37\)	0.479786	−	0.147994i	0.0788763	−	0.0243301i	−0.255066	−	0.966924i	\(-0.582097\pi\)
							0.333942	+	0.942594i	\(0.391621\pi\)
\(41\)	−0.230737	−	3.07897i	−0.0360351	−	0.480854i	−0.985804	−	0.167902i	\(-0.946301\pi\)
							0.949769	−	0.312953i	\(-0.101318\pi\)
\(43\)	0.255416	−	3.40829i	0.0389506	−	0.519760i	−0.943270	−	0.332028i	\(-0.892267\pi\)
							0.982220	−	0.187732i	\(-0.0601137\pi\)
\(47\)	−5.61721	+	7.04376i	−0.819354	+	1.02744i	0.179690	+	0.983723i	\(0.442491\pi\)
							−0.999044	+	0.0437142i	\(0.986081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.353.17	yes	648
9.5	odd	6		441.2.bd.a.59.17	✓	648
49.5	odd	42		441.2.bd.a.299.17	yes	648
441.5	even	42	inner	441.2.bn.a.5.17	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.17	✓	648	9.5	odd	6
441.2.bd.a.299.17	yes	648	49.5	odd	42
441.2.bn.a.5.17	yes	648	441.5	even	42	inner
441.2.bn.a.353.17	yes	648	1.1	even	1	trivial